CMO 2006

Let $f(n, k)$ be the number of ways of distributing $k$ candies to $n$ children so that each child receives at most 2 candies. For example, if $n=3$, then $f(3,7)=0, f(3,6)=1$ and $f(3,4)=6$.

Determine the value of

$$f(2006,1)+f(2006,4)+f(2006,7)+\cdots+f(2006,1000)+f(2006,1003) .$$

Comment. Unfortunately, there was an error in the statement of this problem. It was intended that the sum should continue to $f(2006,4012)$.

Let $A B C$ be an acute-angled triangle. Inscribe a rectangle $D E F G$ in this triangle so that $D$ is on $A B, E$ is on $A C$ and both $F$ and $G$ are on $B C$. Describe the locus of (i.e., the curve occupied by) the intersections of the diagonals of all possible rectangles $D E F G$.

In a rectangular array of nonnegative real numbers with $m$ rows and $n$ columns, each row and each column contains at least one positive element. Moreover, if a row and a column intersect in a positive element, then the sums of their elements are the same. Prove that $m=n$.

Consider a round-robin tournament with $2 n+1$ teams, where each team plays each other team exactly once. We say that three teams $X, Y$ and $Z$, form a cycle triplet if $X$ beats $Y, Y$ beats $Z$, and $Z$ beats $X$. There are no ties.

(a) Determine the minimum number of cycle triplets possible.

(b) Determine the maximum number of cycle triplets possible.

The vertices of a right triangle $A B C$ inscribed in a circle divide the circumference into three arcs. The right angle is at $A$, so that the opposite $\operatorname{arc} B C$ is a semicircle while $\operatorname{arc} A B$ and $\operatorname{arc} A C$ are supplementary. To each of the three arcs, we draw a tangent such that its point of tangency is the midpoint of that portion of the tangent intercepted by the extended lines $A B$ and $A C$. More precisely, the point $D$ on $\operatorname{arc} B C$ is the midpoint of the segment joining the points $D^{\prime}$ and $D^{\prime \prime}$ where the tangent at $D$ intersects the extended lines $A B$ and $A C$. Similarly for $E$ on arc $A C$ and $F$ on arc $A B$.

Prove that triangle $D E F$ is equilateral.